23 Analysis of Kinetics Data

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23ANALYSIS OF KINETICS DATA

In this chapter youll learn how to extract rate constant information fromsimple first-order processes, from biphasic processes and from complex rateprocesses.

EXPERIMENTAL TECHNIQUESIn principle, any measurable property of a reacting system that isproportional to the extent of reaction may be used to monitor the progress of thereaction. The most common techniques are spectrophotometric (UV-visible,fluorescence, IR, polarimetry and NMR) or electrochemical (pH, ion-selectiveelectrodes, conductivity and polarography). Either a batch method can beused, in which samples are withdrawn from the reaction mixture and analyzed,or the reaction may be monitored in situ. By far the most widely used techniqueinvolves UV-visible spectrophotometry.Since reaction rate is sensitive to temperature, the system must be

thermostatted. For most reactions in aqueous solution, the ionic strength shouldbe controlled at a fixed value (see Experimental Techniques in Chapter 22).

ANALYSIS 0~ M~N~PHA~IC KINETICS DATAMost reactions are characterized by a change in reactant or productconcentration that can be described by a single exponential. The differential formof the rate equation contains a single term; the integrated form yields a straightline from which the rate constant can be obtained. Some of the more commonand useful cases are described.

FIRST-ORDER KINETICSFirst-order reactions are by far the most common. They are also the simplestto study experimentally. For reactions of higher order, experimental conditionscan usually be arranged so that they are first-order (see below). This simplifiesthe situation considerably.

373

Excel for Chemists: A Comprehensive Guide. E. Joseph Billo

Copyright 2001 by John Wiley & Sons, Inc.

ISBNs: 0-471-39462-9 (Paperback); 0-471-22058-2 (Electronic)


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374 Excel for Chemists

For the reaction of species A to give product B, with rate constant kkA-----,B

the rate of disappearance of A is proportional to the amount of A:4%- = -k[A]tdt (23-l)

Of course the rate of appearance of product can also be used to monitor thereaction, since-44 t d[Blt--dt - dt

Integration of equation 23-1 leads to the relationshipln Plt = - kt + In [Alo

orlog Mt = -2.303 k t + log [A]0 (23-2)

that is, a plot of the logarithm of the concentration of A, plotted vs. time, yields astraight line from which the rate constant k can be obtained. The intercept term isusuallv of no interest.

An alternative form of equation 23-1 that sometimes is useful is[A]t = [A]oemkt (23-3)

Occasionally a first-order rate constant is obtained by experimentaldetermination of the half-life tin the time required for the reactant concentrationto decrease to one-half of its original value. From equation 23-2 it follows that k =ln(2)/tl/2 = 0.693&/L.If a reaction is monitored by UV-visible spectrophotometry, for example, theconcentration may be replaced by the absorbance (A) in equation 23-2. In thegeneral case, both reactant and product may absorb at the monitoringwavelength, and thus the final absorbance is non-zero. Under these conditionsthe form of equation 23-4 that must be used is

InIAt-A-1 = -kt+hIAi-A-1 (23-4)where Ai is the initial absorbance reading and A, is the absorbance value whenthe reaction is complete. For first-order reactions the rule of thumb is that 10half-lives must elapse before the reaction can be considered to be complete. After10 half-lives a first-order reaction is (1 - 0.51) or 99.9% complete.

Figure 23-l illustrates the application of equation 23-4 in the determination ofthe hydrolysis of a substrate by the enzyme thermolysin. The parameters


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Chapter 23 Analysis of Kinetics Data 375

returned by the SLOPE and INTERCEPT functions were used to calculate thetheoretical line in column D of Figure 23-l. The formula in cell D8 is

=$B$26+A8*$B$25.The first-order behavior is verified by the straight-line fit of the data, shownin Figure 23-2.

AMA] bu Thlermdusin

1.3678 1.292; ..._ I .1 ki1.444 1.1671.437

12 I.20114 1.1581603 1.138.,1 .12

1.9772.2472.517.A............ ...2.787

1 ,2fir:i 1.. .::./ INTERC=EPT= 0.35708031 * .Figure 23-l. Data table for the enzymatic hydrolysis of FAGLA by thermolysin.


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5.0

- 4.0Q):: 3.0E:a 2.0zc- 1.0

I0.0

Hydrolysis of FAGLA

0 10 20time, minutes

Figure 23-2. First-order plot for the hydrolysis of FAGLA.

REVERSIBLE FIRST-ORDER REACTIONSIf the reaction is reversible, e.g.,

kfA -Bkrthen the rate of approach to equilibrium is a first-order process. If the A, valueis denoted by Aq, then the first-order rate expression is simply

WAt-Aeql = -kobsdt+~IAi-AooI (23-5)and kobsd = kf + kr. Only the experimental constant .kobsd can be obtained fromthe first-order plot. If the equilibrium constant is known, the values of kf and krcm be calculated, since kf / k r = Kq.WHEN THE FINAL READING IS UNKNOWN

Occasionally it is not possible to obtain A, - for example, if the reaction isvery slow, if secondary reactions occur toward the end of the primary reaction orif the experiment was terminated before the final reading was obtained.Obviously if a reaction has a half-life of one year it may not be practical to waitfor the reaction to be complete.Several ways have been developed to deal with a reaction for which the A,value is not available. The Guggenheim method, for example, uses paired


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readings at t and t + At to calculate the rate constant. By now you probablyrealize that a much simpler and direct method will be to use the Solver to findboth the rate constant k and the A, value by non-linear least-squares.

The worksheet in Figure 23-3 illustrates a case of a reaction so slow that itwas necessary to use the Solver to find the final absorbance reading. Theunstable cis-octahedral isomer of the nickel(I1) complex of the macrocyclic ligandcyclam (1,4,&U-tetraazacyclotetradecane) isomerizes to the planar complex[Ni(cyclam12+, which absorbs at 450 nm.* In acidic solution the reaction is slow.

Note the use of date and time arithmetic to calculate the elapsed time be-tween readings. The formula in cell B7 is:

=1440*(A7-$A$7)Because the absorbance of the product is being monitored, the formula in cell

07 is:=Af-Af*EXP(-k-obsd*t)The Solver was used to minimize the value in the target cell (El 8, sum of

squares of residuals) by varying the values of the changing cells (Cl 9 and C20,A, and kobsd)-

E. J. Billo, Inorg. Chem. 1984, 23,236.


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SECOND-ORDER KINETICSFor the bimolecular reaction of species A and B to give product or products,with rate constant k kA+BmC

the reaction is second-order and the rate depends on the concentration of both Aand B:dIAlt- = - k [Alt [Bitdt (23-6)

Integration of equation 23-6 yields equation 23-7, which can be used todemonstrate that a reaction is second-order and to obtain the rate constant:1 PI0 [Alt

[Alo - [Blo In [Alo [Blt = kt

2.303 PI0 PutIA10- PI0 log [Alo [B]t = kt (23-7)For the special case [A] = [B], equation 23-7 fails (since the denominator termbecomes zero) and the alternate second-order expression 23-8 must be used:

1 1--EAlt - PI0 - kt (23-8)

The same equation applies if the reaction is second-order in a single reactant, e.g.,d[Al t- =-k[A]t2

dt(23-9)

PSEUDO-FIRST-ORDER KINETICSIf the concentration of species B (for example) is large relative to A, it will

remain essentially unchanged during the course of the reaction, and the rateexpression 23-6 is simplified to 2340, a form of equation 23-1. The reaction issaid to be run under pseudo-@t-order conditions:

d[Alt- - - k fB]i [AIt = -kohsd [AItdt

- (23-10)and thus kobsd = k [B]i (23-l 1)

Once the first-order behavior with respect to [A] has been verified, thereaction can be run with varying concentrations of B (B still in large excess overA). A graph of k,bsd as a function of [B] should be linear; the slope is the rateconstant k. For large variations in [B], resulting in large variations in k,bs& it is


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often useful to plot log k,hsd vs. log [B]. The slope of the plot gives the order ofthe reaction with respect to [B), in this case 1.0.ANALYSIS 0~ BIPHA~I~ KINETICS DATA

Often a plot of concentration vs. time, or the monitored parameter vs. time,or the rate plot, will not be monophasic. This can arise from a number ofdifferent situations, the more common of which are described below.CONCURRENT FIRST-ORDER REACTIONS

If, in a mixture of A and B, these components react by parallel first-orderprocesses to give a common product C, and A and B do not interconvert, then afirst-order plot of the rate of appearance of P will be curved, having a fast and aslow component.ALB&C

This situation is commonly encountered in the measurement of radioactivedecay of a mixture of radioisotopes.CONSECUTIVE FIRST-ORDER REACTIONS

For consecutive first-order processes,

the rate expressions ared[Alt- - - WAltdt -

4Bl t- = k1Alt-k2MtdtWI t- = k2 Witdt

which lead to the following expressions for the concentrations:[Alt = [Alo e-5 t

klPlt = [AIO~ [ eak+ - ewk$ ]

(2342)(2343)


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1 -$+-e-k1 +&eBkzr2- 1 2- 1 1 (23-14)The concentrations of A, B and C for a typical series first-order process are

shown in Figure 23-4.

TimeFigure 23-4. Concentration vs. time for consecutive first-order reactions.

The disappearance of A is purely first-order and can be used to determinethe rate constant kl. The species B is formed and then decays in an unmistakableseries-first-order manner (Figurel2-5 is an example of this). The appearance of Cmay seem to be pure first-order if the slight deviation from first-order behavior atthe beginning of the reaction is missed. In addition, more than one species mayabsorb at a particular wavelength, complicating and confusing the situation. Inthe example that follows, both B and C absorb at the same wavelength. Thisresults in behavior that is similar to, and difficult to distinguish from, concurrentfirst-order reactions.AN EXAMPLE

The unstable cis-octahedral isomer of the nickel(I1) complex of themacrocyclic ligand 13aneNq (l&7,10-tetraazacyclotridecane) isomerizes to anintermediate planar isomer, which then converts to the stable planar isomer


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intermediate planar isomer, which then converts to the stable planar isomer[Ni(13aneN4)2+; both planar isomers absorb at 425 nm.* The reaction exhibits afast and a slow component, as illustrated in Figure 23-5.

The rate constants for the fast and slow reactions can be obtained in thefollowing manner: the rate constant for the slow reaction is obtained from thedata in the latter part of the reaction, by the usual first-order plot. The interceptof this plot at t = 0 is used to obtain A, for the fast reaction; the early-time data isthen used to construct a second first-order plot. The first-order plots of ln(A, -At) vs. t for the data are shown in Figure 5-22.

.7QO

Figure 23-5. Fast (inset) and slow reactions in the isomerization ofcis-[Ni(13aneNq)(H20)2]2+.

* Anne M. Martin, Kenneth J. Grant and E. Joseph Billo, Inorg. Chem. 1986,25,4904.


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Applying LINEST to the data in the straight-line portion of the slow process(rows 17-26 of Figure 23-6) yields the rate constant for the slow process andpermits the calculation of A, from the intercept value (A0 for the slow process isA, for the fast process). From In (A, - Ao) = -1.315, A, - A0 = 0.269, from whichA-= A0 = 0.439, as shown in Figure 23-7.

Figure 23-8. Results of first-order plot of the fast part of the isomerization ofcis-[Ni(13aneNq)(H20)212+.

Having established the value for A,, the data from the early stages of thereaction can be analyzed as a first-order process by plotting In 1At - A - 1 (seethe inset in Figure 5-23). The results are shown in Figure 23-8.CONSECUTIVE REVERSIBLE FIRST-ORDER REACTIONS

The sequence of coupled reversible first-order processes

yields the following set of differential equations:44 t- - - kl [Alt + k2 PI tdt -d[Blt- = kl [AIt - k2 IBIt - k3 [Bl t + k4 [cl tdtd[Clt- = k3 Pit - k4 Kitdt

The differential equations can be solved to yield analytical expressions forthe concentrations of A, B and C. For the case where [B]o = [Cl0 = 0 at t = 0, theexpressions are* : kl CA2 - k3 - k4,& f, kl (k3 + k4 - A31 &3 t

A2 &2- h3) A3 @2 - h3) I (21-15)

John W. Moore and Ralph G. Pearson, Kinetics and Mechanism, 3rd edition, Wiley-Interscience, New York, 1982, p. 298.


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Wit = [Alo kl k4A2A3+ h (k4 - h2) e-h2 t +A2 (A2 - h3)

kl (A3 - k4> e-h3 tA3 (A2 - h3) 1

kl k3 h k3Kit = IAl0 h2 h3 +A2 (A2- h3)[ e-A:! t-h k3

A3 (A2 - h3) e--h3 t 1(23-16)

(23-17)where A2 = (p + qv

A3 = (P-Wp = (kl + k2 + k3+ k4)q = [p2 + 4(klk3 + k2k4 + k1k4)]1/2

(23-18)(23-19)(23-20)(23-21)

A system such as this can readily be solved by using the Solver. Theprevious example of the isomerization of cis-[Ni(13aneN4)(H20)2]2+ (Figure 23-5),is actually a case of coupled reversible first-order processes. There are twoobservable rate processes, kfast and kslow, but each is reversible. Treatment of thedata according to the consecutive reversible first-order scheme is shown inFigure 23-6.

The expressions for [AIt, [B]t , [C]t and Abs,,lc in cells Cl 4, D14, El 4 andF14 are:=C_T*(k-2*k-4/(Ls2*L-3)+k-1 *(L-2-k-3-k-4)*EXP(-LB2*t)/L-2*(L-2-L-3))+k-l *(k-3+k-4-L_3)*EXP(-L_3*t)/

(L3*(L2-L3)))=C-T*(kJ *k-4/(L-2*L-3)+k-l *(k-4-L-2)*EXP(-L_2*t)/(L-2*(L-2-L-3))+k-l *(L-3-k-4)*EXP(-L-3*t)/(L-3*(L-2-L-3)))=C-T*(kJ *k~3/(L~2*L~3)+k~l *k~3*EXP(-L~2*t)/(L~2*(L~2-L~3))-k-1 *k-3*EXP(-L-3*t)/(L-3*(L-2-L-3)))

The quantities p, q, h2 and h3 were calculated by using the formulas:

=k l+k 2+k 3+k 4- - - -=SQRT(P*P-4*(k-1 *k-3+k-2*k-4+k-l k-4))=( P+Q)/2=( P-Q)/2


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The Solver was used to minimize the sum of squares of residuals (cell G12,Figure 23-9) by changing the values of kl, k2, k3 and k4, and the molarabsorptivity of the intermediate species B (EA and EC could be determinedseparately). The Solver provided a reasonable fit to the data. All five changingcells could be varied at once, although approximate estimates of the parametershad to be provided before the Solver could proceed to a solution. The fiveparameters of the Solver solution are shown in bold in Figure 23-9.


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386

SIMULATION OF KINETICSBY NUMERICAL INTEGRATION

Excel for Chemists

Complex systems like that of Figure 23-5 can also be analyzed by Runge-Kutta numerical integration discussed in Chapter 9. The big advantage of directnumerical integration is its generality. The method can be applied to reactionsequences of (in theory, at least) any complexity. The following example appliesthe Runge-Kutta method to a more complex case.In the experiment described in the preceding section, the reactant species Awas prepared from the final product C. The absorbance of the experiment at t = 0showed that the solution contained a small amount of unconverted C. Instead of[A]0 = 1.43 mM and [B]o = [C]o = 0, the initial conditions were actually [A]0 =

1.37 mM, [B]o = 0, [Cl0 = 0.06 mM. This set of initial condition cannot beaccurately treated by using equations 23-15,23-16 and 23-17, but it can be treatedby using RK integration.The worksheet of Figure 23-9 was modified to use the RK method, asdescribed in Chapter 9.The Runge-Kutta formulas entered in row 18, for TAl, TA2, TA3, TA4 and

[AIt+&, and for TBl, etc, are:=(-k-l *C_A+k_2*C_B)*(Al g-Al 8)=(-k-l *(C~A+TA1/2)+k~2*(C~B+TB1/2))*(A19-A18)=(-k-l *(C-A+TA2/2)+k-2*(C-B+TB2/2))*(A19-A18)=(-k-l *(C-A+TA3)+k-2*(C_B+TB3))*(Al g-Al 8)=CAA+(TA1+2*TA2+2*TA3+TA4)/6=(k-1 *C-A-(k_2+k_3)*C_B+k_4*C_C)*(Al g-Al 8)=(kJ *(C_A+TA1/2)-(k_2+k_3)*(C_B+TB1/2)

+k_4*(C_C+TC1/2))*(Al g-Al 8)=(k-1 *(CmA+TA2/2)-(k-2+k-3)*(C_B+TB2/2)

+k-4*(CsC+TC2/2))*(A19-A18)=(k-1 *(CmA+TA3)-(k-2+k-3)*(C_B+TB3) +k-4*(CeC+TC3))*(A19-Al 8)=C B+(TB1+2*TB2+2*TB3+TB4)/6-

=(k_3*(C_B+TB2/2)-k_4*(C_C+TC2/2))*(Al g-Al 8)=(k-3*(C-B+TB3)-ks4*(C_C+TC3))*(A1 g-Al 8)=CsC+(TC1+2*TC2+2*TC3+TC4)/6


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Following the layout suggested in Chapter 9, the initial concentration of Awas entered in cell Cl 8; cell Cl 9 contains the formula =H 18. Similar formulaswere used for [B]t and [C]t.

When the RK method (or any method involving numerical integration) isused, it is important to chose time increments small enough to assure accuracy incalculations. In the RK formulas above, a calculated time increment, e.g. (A19-A18) in row 18, was used rather than a constant value, so that the intervalbetween successive data points could be varied. In this way larger timeincrements can be used at the end of the reaction, when concentrations arechanging slowly. Only a few cells in column B contain A&d measurements;only one is shown in the spreadsheet fragment of Figure 23-10. The data table oftime and A,bsd values was located elsewhere in the worksheet; VLOOKUP wasused in column B to enter the A,bsd values for the appropriate time values. Theformula is as follows:

=IF(ISERROR(VLOOKUP(t,DataTable,l ,O)),,VLOOKUP(t,DataTable,2,0))The ISERROR function was used with range-lookup = 0; otherwise VLOOKUPreturns #N/A! for all values of t for which there is no corresponding entry inDataTable.An IF statement was used to calculate the squares of residuals only for rowsthat contained an A&d value.

The Solver was used in the same way as in the preceding example, althoughit was necessary to use subsets of the complete set of parameters to perform theinitial stages of the refinement. Note the difference in the results when the smallamount of product species that was present at the beginning of the reaction istaken into acount: the values of the regression parameters are significantlydifferent, and the sum of squares of residuals (1.38 x lOA) is significantly smallerthan in the preceding treatment (4.52 x 104).Calculation time will be significantly longer for a solution by direct

numerical integration than for a solution using analytical expressions.


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k-l = : 4,34E-04

small 3mountofC present: [C]i=

Moe .f%EA=E-b=,.. .,. ,.,,.,.

. . ..E*=... ...

.,1.37E-03'..,. .. ,.,.,I6 .##E-05

+A2 .-5.92E-06-5.87E-06:=j ,#iE-&-5.77E-#i-5.72E-06-5.6%0i-5.62E-06-5.57E-06-l.l#E-05-l.WE-05-l.WE-05- j, .04ET05-2.03E-05-1.96E-05

Figure 23-10. Data table for the isomerization of cis-[Ni(13aneNq)(H20)2]2+ analyzed bythe RK method. Columns I through Y (not shown) contain formulas forcalculation of [B] and [Cl.

23 Analysis of Kinetics Data

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